Linear inequalities in one unknown · Ages 14–15
Inequalities and the number line
> < hollow, ≥ ≤ filled; greater runs right, less runs left.
How to do it
Method
> < hollow, ≥ ≤ filled; greater runs right, less runs left
x ≥ −2: a filled circle at −2, running right
Another way
Put a number in
x ≥ −2: −2 ≥ −2 is true, so −2 is a solution
In full
There are four inequality signs: > greater than, < less than, ≥ greater than or equal to (not less than, at least), ≤ less than or equal to (not greater than, at most, not more than). The open side faces the bigger number: 5 > 3, −4 < 1. x > 3 means every number bigger than 3: 3.5, 4 and 100 are all solutions, endlessly many, but 3 itself isn't. On a number line: draw a circle at 3, hollow when 3 is left out (> or <), filled when it is in (≥ or ≤); greater runs to the right, less to the left. Or check by putting a number in: any number that makes the inequality true is a solution. x ≥ −2: put in −2, and −2 ≥ −2 is true, so −2 is a solution and the circle is filled.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
How is "x is at most −10" written as an inequality?
- A. x ≥ −10
- B. x > −10
- C. x < −10
- D. x ≤ −10
Choose the right answer
Show the answer and working
Answer: D. x ≤ −10
"At most", "not greater than" and "not over" are all ≤: equal to −10 or less.
Example 2
x is a positive whole number and x < 16. How many values can x take?
Show the answer and working
Answer: 15
Positive whole numbers start at 1, and x < 16 leaves 16 out: 1, 2, …, 15, which is 15 of them.
Example 3
The number line shows the solutions of an inequality. The least whole-number solution is
Show the answer and working
Answer: −3
The line runs right from −4: filled, −4 itself is a solution; hollow, −4 is left out and the least whole number is the next one. Here it is −3.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.How is "x is at most 10" written as an inequality?
- A. x < 10
- B. x > 10
- C. x ≥ 10
- D. x ≤ 10
Choose the right answer
Answer:
2.Which inequality does the number line show?
- A. x ≤ 5
- B. x ≥ 5
- C. x < 5
- D. x > 5
Choose the right answer
Answer:
3.Which number line shows x ≤ −5?
- A
- B
- C
- D
Choose the right picture
Answer:
4.How is "x is not less than 6" written as an inequality?
- A. x > 6
- B. x ≤ 6
- C. x ≥ 6
- D. x < 6
Choose the right answer
Answer:
5.The number line shows the solutions of an inequality. The least whole-number solution is
Answer:
All the answers
- 1. D. x ≤ 10
- 2. A. x ≤ 5
- 3. D
- 4. C. x ≥ 6
- 5. 7
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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